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Neostability transfers in derivation-like theories

Omar Leon Sanchez, Shezad Mohamed

TL;DR

This work develops a general framework of derivation-like expansions of base theories and proves that, when a model companion exists, key model-theoretic properties—such as completeness, quantifier elimination, and neostability notions (stability, simplicity, NSOP$_1$, rosiness)—transfer from the base theory $T_0$ to the companion $T_+$. Central to the approach is an induced independence concept $\mid^+_C$ built from $\operatorname{acl}_+$ and the base relation $\mid\smile^0$, enabling Kim-Pillay–style transfer arguments. The paper then provides a broad array of concrete instances (e.g., separably closed fields with Hasse–Schmidt structures, $\mathcal D$-fields in characteristic zero, differential fields in positive characteristic, CCMs with meromorphic vector fields, and theories with automorphisms) where the transfer phenomena hold, yielding stability, simplicity, NSOP$_1$, and rosiness results for the model companions. Overall, the results supply a unified, operator-theoretic toolkit for establishing tame model-theoretic behavior across diverse derivation-like systems.

Abstract

Motivated by structural properties of differential field extensions, we introduce the notion of a theory $T$ being derivation-like with respect to another model complete theory $T_0$. We prove that when $T$ admits a model companion $T_+$, several model-theoretic properties transfer from $T_0$ to $T_+$. These properties include completeness, quantifier elimination, stability, simplicity, and NSOP$_1$. We also observe that, aside from the theory of differential fields, examples of derivation-like theories are plentiful.

Neostability transfers in derivation-like theories

TL;DR

This work develops a general framework of derivation-like expansions of base theories and proves that, when a model companion exists, key model-theoretic properties—such as completeness, quantifier elimination, and neostability notions (stability, simplicity, NSOP, rosiness)—transfer from the base theory to the companion . Central to the approach is an induced independence concept built from and the base relation , enabling Kim-Pillay–style transfer arguments. The paper then provides a broad array of concrete instances (e.g., separably closed fields with Hasse–Schmidt structures, -fields in characteristic zero, differential fields in positive characteristic, CCMs with meromorphic vector fields, and theories with automorphisms) where the transfer phenomena hold, yielding stability, simplicity, NSOP, and rosiness results for the model companions. Overall, the results supply a unified, operator-theoretic toolkit for establishing tame model-theoretic behavior across diverse derivation-like systems.

Abstract

Motivated by structural properties of differential field extensions, we introduce the notion of a theory being derivation-like with respect to another model complete theory . We prove that when admits a model companion , several model-theoretic properties transfer from to . These properties include completeness, quantifier elimination, stability, simplicity, and NSOP. We also observe that, aside from the theory of differential fields, examples of derivation-like theories are plentiful.
Paper Structure (9 sections, 24 theorems, 37 equations)

This paper contains 9 sections, 24 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Examples
  5. Separably closed fields and Hasse-Schmidt fields
  6. $\mathcal{D}$-fields in characteristic zero
  7. Differential fields in positive characteristic
  8. CCMs with meromorphic vector fields
  9. Theories with an automorphism

Key Result

Theorem 1.1

Suppose $\mathop{\hbox{$\mid$} \hbox{$\smile$}}^0$ is nonforking independence in $T_0$ and that for every model $M \models T_+$ we have that $\operatorname{dcl}_0(M) \models T_0$. Suppose also that $T$ is derivation-like with respect to $(T_0, \mathop{\hbox{$\mid$} \hbox{$\smile$}}^0)$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 3.1
  • Remark 3.2
  • ...and 45 more